The equation of a hanging cable (for nearly any loading condition) is easily derivable. A good text book in analytical or classical mechanics will have a discussion and solution. My text "Mechanics", 3rd edtion, by Symon is one such book. The section on cables is only 3 pages long, and the explanation is complete.
I have also seen the solution derived from entropy considerations.
Loaded bridge cables are not always catenaries.
The Golden Gate Bridge is Parabolic!
SILICONE IS FAKE--DO THE MATH.